The terminology Kalman filter, as defined in Wikipedia, also known as linear quadratic estimator (LQE), refers to an algorithm that uses a series of measurements observed over time, containing noise (random variations) and other inaccuracies, and produces estimates of unknown variables that tend to be more precise than those based on a single measurement alone. The Kalman filter operates recursively on streams of noisy input data to produce a statistically optimal estimate of the underlying system state. Applications of the Kalman filter include guidance, navigation and control of vehicles, particularly aircraft and spacecraft, and time series analysis used in fields such as signal processing and econometrics.
The algorithm works in a two-step process. In the prediction step, the Kalman filter produces estimates of the current state variables, along with their uncertainties. Once the outcome of the next measurement (necessarily corrupted with some amount of error, including random noise) is observed, these estimates are updated using a weighted average, with more weight being given to estimates with higher certainty. Because of the algorithm's recursive nature, it can run in real time using only the present input measurements and the previously calculated state; no additional past information is required.
In general, estimation process deals with recovering the desired variables of a dynamic system from available measurements. Though estimators such as the Kalman filter has been successfully used in many engineering problems, it is based on assumptions that the dynamic model of the system under consideration is precisely known and the system dynamics/measurements are subjected to Gaussian processes with known statistics. Even though the Kalman filter possesses some innate robust characteristics, the estimator performance degradation in the presence of system uncertainties may not be tolerable. There exist four main approaches to deal with the robust estimation problem and they are based on i) H∞ filtering, ii) setvalued estimation, iii) guaranteed-cost filtering, and iv) regularized least-squares. All these approaches are compared in A. H. Sayed, “A framework for state-space estimation with uncertain models,” Automatic Control, IEEE Transactions on, vol. 46, no. 7, pp. 998-1013 (July 2001), where relevant concerns on parameterizations, stability, robustness, and online implementation of each approach are addressed. None of these existing schemes can guarantee asymptotic convergence of the mean estimation error in the presence of persistent excitation or disturbance that is not asymptotically decaying to zero. Most of the current robust Kalman filter schemes only consider minor system parametric uncertainties or asymptotically decaying disturbance, if the disturbance is non-zero mean. Therefore, the estimates obtained using the existing schemes would yield biased estimates in the presence of persistent excitation. In H∞ filtering, estimators are designed to minimize the worst case H∞ norm of the transfer function from the noise inputs to the estimation error output. See for example, A. H. Sayed, “A Framework For State-Space Estimation With Uncertain Models,” Automatic Control, IEEE Transactions on, Vol. 46, No. 7, pp 998-1013, (July 2001), de Souza, C. E., et al., “H∞ Estimation For Discrete Time Linear Uncertain Systems,” International Journal of Robust and Nonlinear Control, Vol. 1, pp. 11-23, (1991). Xie, L., et al., “H∞ Filtering For Linear Periodic Systems With Parameter Uncertainty,” Systems & Control Letters, Vol. 17, pp. 343-350, (1991), Nagpal, K. et al., “Filtering and smoothing in an H∞ setting,” Automatic Control, IEEE Transactions on, Vol. 36, No. 2, pp. 152-166, (February 1991), Fu, M., et al. “H∞ estimation for uncertain systems,” International Journal of Robust and Nonlinear Control, Vol. 2, pp. 87-105, (1992), Bernstein, D. S. et al., “Mixed-norm H2/H∞, Regulation and Estimation: The Discrete-time Case,” Systems & Control Letters, vol. 1.6, pp. 235-248 (1991), and Wang, Z., et al., “Robust H2/H∞^state Estimation For Systems With Error Variance Constraints: The Continuous-time Case,” Automatic Control, IEEE Transactions on, vol. 44, no. 5, pp. 1061-1065 (May 1999). Since H∞ filtering is a worst-case design method, while guaranteeing the worst-case performance, it generally sacrifices the average filter performance. A robust estimator design methodology based on iteratively solving a tradeoff problem between estimator performance and robustness is presented Xu, H., et al., “A Kalman Filter Design Based on the Performance/Robustness Tradeoff” Journal of Latex Class Files, Vol. 1, No. 11, pages 1-7 (November 2002). A robust state estimator for a class of uncertain systems where the noise and uncertainty are modeled deterministically via an integral quadratic constraint is presented in Savkin, A, et al. “Recursive State Estimation for Uncertain Systems with an Integral Quadratic Constraint,” IEEE Transactions on Automatic Control, Vol. 40. No. 6, page 1080 (June 1995). This approach, known as the set-valued state estimation (see Bertsekas., et al. “Recursive State Estimation for a Set-Membership Description of Uncertainty”˜EEE Transactions On Automatic Control, AC-16, No. 2, page 117 (April 1971) involves finding the set of all states consistent with given output measurements for a system with norm bounded noise input (see James, “Nonlinear State Estimation for Uncertain Systems with an Integral Constraint” IEEE Transactions on Signal Processing, Vol. 46, No. 11, page 2916 (November 1998). Robust estimation approach known as the guaranteed cost filtering is presented in Petersen, I., et al., “Robust State Estimation for Uncertain Systems,” Proceedings of the 30th Conference on Decision and Control, F2-7-2:50, IEEE page 2630, Brighton, England, (December 1971), Xie, L., et al. “Robust Kalman Filtering for Uncertain Discrete-Time Systems” IEEE Transactions on Automatic Control, Vol. 39, No. 6, page 1310 (June 1994), and Petersen, I. et al. “Optimal Guaranteed Cost Control and Filtering for Uncertain Linear Systems” IEEE Transactions on Automatic Control, Vol. 39, No. 9, page 1971, (September 1994). Here, estimators are designed to guarantee that the steady-state variance of the state estimation error is upper bounded by a certain constant value for all admissible model uncertainties. Design of robust estimators that ensure minimum filtering error variance bounds for systems with parametric uncertainty residing in a polytope are given in Shaked, U., et al. “New Approaches to Robust Minimum Variance Filter Design” IEEE Transactions on Signal Processing, Vol. 49, No. 11, page 2620. (November 2001) and de Souza, C., et al., “Robust Filtering for Linear Systems With Convex-Bounded Uncertain Time-Varying Parameters,” IEEE Transactions on Automatic Control, Vol. 52, No. 6, page 1132, (June 2007). While the H∞ formulation and the guaranteed cost filtering involves de-regularization, a robust estimator design based on the regularized least-squares approach is presented in Sayed, Ali H. “A Framework for State-Space Estimation with Uncertain Models” IEEE Transactions on Automatic Control, Vol. 46, No. 7, Page 998, (July 2001). Extension of the regularized least squares approach to time-delay systems and time varying system are presented in Subramanian, A., et al. “Multiobjective Filter Design for Uncertain Stochastic Time-Delay Systems” IEEE Transactions on Automatic Control, Vol. 49, No. 1, Page 149, (January 2004) and Subramanian, A., et al., “Regularized Robust Filters for Time-Varying Uncertain Discrete-Time Systems” IEEE Transactions On Automatic Control, Vol. 49, No. 6, Page 970 (June 2004), respectively. Petersen and Savkin provides a comprehensive research monograph on robust filtering for both discrete and continuous time systems from a deterministic as well as H∞ point of view I. R. Petersen and A. V. Savkin. “Robust Kalman Filtering For Signals and Systems with Large Uncertainties.” Boston, Mass.: Birkhauser, 1999. More recently, Neveux, P., et al., “Robust Filtering for Linear Time-Invariant Continuous Systems” IEEE Transactions On Signal Processing, Vol. 55. No. 10, Page 4752 (October 2007) and Zhou, Tong, “Sensitivity Penalization Based Robust State Estimation for Uncertain Linear Systems,” IEEE Transactions on Automatic Control, Vol. 55, No. 4, Page 1018 (April 2010) propose a robust filtering approach based on penalizing the sensitivity of estimation errors to parameter variations. Furthermore, the optimal joint filtering and parameter identification problem for uncertain linear stochastic systems with unknown parameters in both state and observation equations is tackled in Basin, M., et al., “Optimal Filtering for Uncertain Linear Stochastic Systems,” Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P. R. China, ThA07.4, page 3376 (Dec. 16-18, 2009)